Theorem redcwlpo 14663

Description: Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse.

Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 14662). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones.

Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO".

WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10242 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)

Assertion

Ref	Expression
redcwlpo	|- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )

Distinct variable group:   x, y

Proof of Theorem redcwlpo

Dummy variables f i j z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> A. x e. RR A. y e. RR DECID x = y )
2		elmapi 6667	. . . . . . . . 9 |- ( f e. ( { 0 , 1 } ^m NN ) -> f : NN --> { 0 , 1 } )
3	2	adantl 277	. . . . . . . 8 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> f : NN --> { 0 , 1 } )
4		oveq2 5880	. . . . . . . . . . 11 |- ( i = j -> ( 2 ^ i ) = ( 2 ^ j ) )
5	4	oveq2d 5888	. . . . . . . . . 10 |- ( i = j -> ( 1 / ( 2 ^ i ) ) = ( 1 / ( 2 ^ j ) ) )
6		fveq2 5514	. . . . . . . . . 10 |- ( i = j -> ( f ` i ) = ( f ` j ) )
7	5, 6	oveq12d 5890	. . . . . . . . 9 |- ( i = j -> ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = ( ( 1 / ( 2 ^ j ) ) x. ( f ` j ) ) )
8	7	cbvsumv 11362	. . . . . . . 8 |- sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = sum_ j e. NN ( ( 1 / ( 2 ^ j ) ) x. ( f ` j ) )
9	3, 8	trilpolemcl 14645	. . . . . . 7 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) e. RR )
10		1red 7969	. . . . . . 7 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> 1 e. RR )
11		eqeq1 2184	. . . . . . . . 9 |- ( x = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) -> ( x = y <-> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y ) )
12	11	dcbid 838	. . . . . . . 8 |- ( x = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) -> (DECID x = y <-> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y ) )
13		eqeq2 2187	. . . . . . . . 9 |- ( y = 1 -> ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y <-> sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
14	13	dcbid 838	. . . . . . . 8 |- ( y = 1 -> (DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = y <-> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
15	12, 14	rspc2v 2854	. . . . . . 7 |- ( ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) e. RR /\ 1 e. RR ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
16	9, 10, 15	syl2anc 411	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> ( A. x e. RR A. y e. RR DECID x = y -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 ) )
17	1, 16	mpd 13	. . . . 5 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 )
18	3, 8	redcwlpolemeq1 14662	. . . . . 6 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> ( sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 <-> A. z e. NN ( f ` z ) = 1 ) )
19	18	dcbid 838	. . . . 5 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> (DECID sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( f ` i ) ) = 1 <-> DECID A. z e. NN ( f ` z ) = 1 ) )
20	17, 19	mpbid 147	. . . 4 |- ( ( A. x e. RR A. y e. RR DECID x = y /\ f e. ( { 0 , 1 } ^m NN ) ) -> DECID A. z e. NN ( f ` z ) = 1 )
21	20	ralrimiva 2550	. . 3 |- ( A. x e. RR A. y e. RR DECID x = y -> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 )
22		nnex 8921	. . . 4 |- NN e. _V
23		iswomninn 14658	. . . 4 |- ( NN e. _V -> ( NN e. WOmni <-> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 ) )
24	22, 23	ax-mp 5	. . 3 |- ( NN e. WOmni <-> A. f e. ( { 0 , 1 } ^m NN )DECID A. z e. NN ( f ` z ) = 1 )
25	21, 24	sylibr 134	. 2 |- ( A. x e. RR A. y e. RR DECID x = y -> NN e. WOmni )
26		nnenom 10429	. . 3 |- NN ~~ om
27		enwomni 7165	. . 3 |- ( NN ~~ om -> ( NN e. WOmni <-> om e. WOmni ) )
28	26, 27	ax-mp 5	. 2 |- ( NN e. WOmni <-> om e. WOmni )
29	25, 28	sylib 122	1 |- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 834   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2737   {cpr 3593   class class class wbr 4002   omcom 4588   -->wf 5211   ` cfv 5215  (class class class)co 5872   ^m cmap 6645   ~~ cen 6735  WOmnicwomni 7158   RRcr 7807   0cc0 7808   1c1 7809   x. cmul 7813   / cdiv 8625   NNcn 8915   2c2 8966   ^cexp 10514   sum_csu 11354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-mulrcl 7907  ax-addcom 7908  ax-mulcom 7909  ax-addass 7910  ax-mulass 7911  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-1rid 7915  ax-0id 7916  ax-rnegex 7917  ax-precex 7918  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924  ax-pre-mulgt0 7925  ax-pre-mulext 7926  ax-arch 7927  ax-caucvg 7928

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-isom 5224  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-recs 6303  df-irdg 6368  df-frec 6389  df-1o 6414  df-2o 6415  df-oadd 6418  df-er 6532  df-map 6647  df-en 6738  df-dom 6739  df-fin 6740  df-womni 7159  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-reap 8528  df-ap 8535  df-div 8626  df-inn 8916  df-2 8974  df-3 8975  df-4 8976  df-n0 9173  df-z 9250  df-uz 9525  df-q 9616  df-rp 9650  df-ico 9890  df-fz 10005  df-fzo 10138  df-seqfrec 10441  df-exp 10515  df-ihash 10749  df-cj 10844  df-re 10845  df-im 10846  df-rsqrt 11000  df-abs 11001  df-clim 11280  df-sumdc 11355

This theorem is referenced by: (None)